The Skorokhod reflection problem for functions with discontinuities (contractive case)

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1 The Skorokhod reflection problem for functions with discontinuities (contractive case) TAKIS KONSTANTOPOULOS Univ. of Texas at Austin Revised March 1999 Abstract Basic properties of the Skorokhod reflection problem for discontinuous functions (but rightcontinuous with left limits) are collected in these notes. In particular, we are concerned with existence and uniqueness in dimensions when the reflection matrix is of contractive type. Some proofs are new, while others are modifications and extensions of existing proofs for the more well-known reflection problem for continuous functions. The article has an overview character, in that the results and proofs are self-contained and are presented fully even in the simplest case of dimension one. Some variations in the proofs, resulting in short and intuitive arguments, reflect the author s own taste. We are also concerned with continuity and discontinuity properties of the reflection mapping under a variety of topologies. The reflection mapping has become a standard tool in numerous areas and the scope of this article is to provide a rigorous and accessible review of part of the subject. 1 Introduction: the physics of the problem The Skorokhod reflection problem is defined in a domain of, which is assumed to be a closed convex set with nontrivial interior. Associated to the boundary is a cone field, i.e., an assignment of a cone to each point of. This is all that is needed for the definition of the problem. Physically, what happens can be described as follows. We are given a free or unconstrained motion "!#!%$'&(, which is a function from ) &*+, into. The function is taken to be continuous or, more generally, right-continuous with left limits at all points of time, i.e. an element of the space - ) &*+,. The problem is to find a function./0.1"!##!2$3&(, of locally bounded variation, such that "!546.1"!# belongs to, for all times!, and, what is more important,. should be chosen so that it is minimal 1 with respect to the cone field. Historically, the problem was first formulated for,7) &*+, in [Skorokhod 1961] as a means of pathwise construction of a solution to a stochastic differential equation with reflection. The physical description of this section can be 8 takis@alea.ece.utexas.edu 1 in some sense to be made precise later 1

2 2 felt in the paper of [Tanaka 1979], where the cone field consists the normal cones at the boundary points of a convex region. If "! happens to be in for all!, then. is identically zero. Otherwise,. should increase only at those times! for which the constrained motion "! 4.1"!# belongs to the boundary. Now, this increase should be consistent with the cone field, i.e., if at time!, "!54.1"!#, then the increase of. at the same time should be contained in the cone " "!#146."!#. The above physical definition can be made precise. We are not going to that in its generality but only work with some special, yet important, cases. We refer to the above problem (i.e. the rigorously defined version of it) as the SRP in with cone field. In many cases, the more restricted version of it, where the free motions are supposed to be continuous, is of importance. By solution to the problem we mean finding the function. or the resulting motion 4.. There are many questions that arise. First, it is clear that not all cone fields make sense. For example, if the cone points toward the exterior of at some point, then it should be intuitively clear that there is no solution for a particularly chosen free process. On the other hand, even if the cone points toward the interior, we are still not sure whether we have 0, 1 or multiple solutions. To date, we do not know much about existence and uniqueness of the SRP in general. However, we know how to give precise existence conditions for the SRP in the positive orthant. Regarding uniqueness, not much is known, even in this special case. One canonical case which is well-studied is what we will call the contractive case. This refers to a SRP in, the closed positive orthant, with specially chosen cone field. Our purpose is to review this case in detail, starting from the trivial case of dimension 1, and provide proofs for free motions in - ) &*+,. These are extensions of existing proofs for continuous free motions. After presenting the proofs, we will deal with some continuity results and point out some discontinuities of the reflection mapping. 2 Dimension one A closed convex domain here is either a compact interval ), or an interval of the type ) +, or +. We shall consider only the case ) &*+,. Associated with the boundary point &( is a trivial cone that either points down or up. In the former case, it is easy to see that the SRP makes no sense. As before, - ) &*+, is the set of right-continuous with left limits functions with real values, while - ) &*+ is the set of functions in - ) &*+, which are increasing 2. Rigorously then, the problem can be cast as follows. - ). ) 4. & Definition 1 (Standard formulation of the SRP). Given a function &*+,, we say that - &*+, is a solution to the SRP iff (i) $,&, (ii) 1 "!#,&.1"!#. Let us note that, by convention, we set.1& &, and the integrator. in the above Lebesgue- Stieltjes integral assigns mass.& 7.1& at the point! &. That the problem has a unique 2 increasing is used in the sense of non-decreasing

3 3 solution is well-known. We repeat the steps that lead to the proof of this fact, as well as the standard formula. Theorem 1. The SRP as specified in Definition 1 has a unique solution. Proof. Let - ) &*+, be given. Note that we will not assume that & &. Any real value for & is possible. By convention, we let & &. Same convention applies to all elements of - ) &*+. Suppose that. is a solution (if any) of the SRP of Definition 1. Then, for any &!, Putting the above together we arrive at 3 :."! $.1 $ because $,&*."! $,&* since, by assumption,. is non-negative.."! $ #& (1) We thus proved that if there is a solution then (1) should hold. The right hand side of (1) is important so we give it a name: (2). "! ) & Note that we only used requirement (i) to derive this. Next we show that. is a solution itself. It is immediate that. is increasing and that 4. $,&. So (i) is satisfied. Let 4.. It remains to show (ii), i.e. that 1 "!,&. "! &. We claim that if "! &, then there is &, such that. "! 4. "!. The claim follows from right continuity: Given any &, we can find &, such that "!#, for all /)!#!4!. Since "!,&, we can choose &"#$", "!#. So we have that, for all )!#!14%, 6 "!# % "!# / "!#. "!. Hence, from (2), Again from (2), & ('. "!14# )+ (' & ) &,-. ) * & (3) & (' Due to (3), the last term can be dropped, and hence. "! 4/ ) &01.. But the latter quantity is, due to (2),. "!. Thus, for all! $ &, "! & implies that. "! 4. "! and so the integral in (ii) is zero. We have constructed a solution given by formula (2). For this solution we have "! ) "! "!2 The above proof is not the only one. One can devise other proofs of existence, perhaps more sophisticated and shorter. But the proof given is, at least, very mundane. 3 where 3 stands for maximum and 4 for minimum

4 4 What about uniqueness? Again there are many ways to prove that the starred solution above is the only one. We adopt the most straightforward one. We first argue that any solution must start from 4.1& & 0 This actually follows easily from (ii). According to our convention (stated just after Definition 1), & &.1& " & 46.& #.1& 2 If & & then.& & and so.1& &. If & &, then & 4.& &, and there is no choice other than.& &. The only case left is & "3&. But we know that.1& $'. & 2$ &. So.1&,&. Thus the first term of the display above becomes null:.1& 4 & &. Suppose then that. is another solution, different from.. From what we showed earlier,. can only be larger than or equal to., while.1&. &. Hence our assumption that. is distinct from. can only mean that. is strictly larger than. at some point. Since. and. start from the same value, we must have that.. for some & and for all sufficiently small. (4) But then.1. $. So.1 4 '&. From (ii) of Definition 1 we deduce the existence of some (small) positive for which.1 4%.1 and this contradicts (4). Hence. is unique. There is another formulation of the SRP which is equally useful. It is the following: Definition 2 (Monotonic formulation of the SRP). Given - ) &*+,, consider the set. ) &*+ %46. $&(, equipped with the componentwise partial ordering. A solution to the SRP - is defined as a minimal element of this set. Our previous discussion showed the following: Theorem 2. The SRP as specified in Definition 2 has a unique solution that coincides with.. This definition is more general than the previous one, because it does not require that have any structure. Indeed, we need not specify that is an element of - ) &*+,. Any 1) &*+ can be used. Similarly, the topology of the domain ) &*+, is irrelevant: only the order structure is needed. Under such generalizations, the problem of Definition 2 makes sense and Theorem 2 holds. 3 Continuity properties We now view the SRP as defining a mapping from - ) &*+, into itself, associating to an -) &*+, the function. of (2): "!# ) & 4 here stands for ; we also use for (5)

5 5 We also let "!# "!54 "!# ) ) "!# "!2 (6) The point of this section is to show that and are continuous under any reasonable topology on - ) &*+. Two topologies are considered in this section: the -topology, or topology of uniform convergence on compact sets, and the -topology introduced in [Skorokhod 1961]. A third topology will be considered in a later section. To start, note that (5), (6) imply that if is continuous then, are also continuous functions. The trivial inequality % where, are two collections of real numbers indexed by the same set, is responsible for the inequalities & "!# "!# & "! "! & "!# "!# (7) & "!# "! & "!# "!# # - ) for any,&. Thus, if we equip &*+, with the -topology, which is induced by the norm we have implying that Recall now that the is continuous under this topology. & (8) topology (the so-called usual Skorokhod topology): Let be the set of strict time changes of ) &*+,, i.e. the set of continuous strictly increasing functions ) &*+ ) &*+,, with & 3& and +, 7+. We say that such that -converges to the identity (id), and -converges to. It is clear that " #! id -converges to iff there is a sequence (9) is a metric for the topology. To see that and are -continuous (i.e. when both the domain and codomain are given the topology) observe the following simple lemma: Lemma 1. For any, and any,- ) &*+,, we have ", and " 1. Proof. Direct application of formula (5) for and (6) for. Theorem 3. Both and are -continuous.

6 6 Proof. Suppose that -converges to. Then there is, -converging to id, such that -converges to. Then, for the same sequence, we have ". But -converges to (composition is -continuous), and is -continuous, so -converges to. Similar argument holds for. The reason that we discussed the and topologies now, is that we are going to use them to prove existence and uniqueness in many dimensions. One more point should be stressed. The space -) &*+ is not complete under the metric of (9). The reason is that measuring the deviation of from id in a uniform fashion is too crude. To make the space complete we need to measure the deviation of from id by measuring the deviations of its slopes, in a sense; see [Billingsley 1968, pg. 112]. So we introduce the norm on by and then set It can be shown that that " #.&. "!#! %, generate the same topology, and that " # This obvious inequality will be useful in the next section. 4 Multidimensional case 2 (10) is complete. Furthermore, notice (11) We consider a very canonical instance of the SRP in. The boundary is the union of the -dimensional faces &( The relative interior (with respect to ) of face is the set 5 &*#,& for all # Given vectors (or, rather, directions) # each associate the positive cone spanned by those directions for which in, we defined the cone field as follows: To we associate the positive cone (half line) generated by. To any other, we. Formally then, "1 1" & $,& Thus, to the origin &, we associate the cone spanned by all vectors. The vectors also called reflection directions. The reflection matrix is a matrix. # # are whose columns are the vectors

7 7 To say that a (locally bounded variation) function "!#!%$3&(, with values in increases, at time!, in a direction contained in the cone spanned by #, is to say that there are nonnegative numbers. "! #. "!, not all zero, such that "!. "! "!#. This simple observation, along with the informal definition of the SRP given in the Introduction, motivates the following: Definition 3 (Standard formulation in ). Given a matrix and - ) &*+, we say that. - ) &*+, is a solution to the SRP( ) iff (i) 4. $,&, (ii) 1 "!,&.#"!# &,. We will use the notation SRP( ) for this problem. As in dimension 1, we want to consider conditions that ensure existence and uniqueness. A word of caution: when we say uniqueness we may mean uniqueness of the reflector. or uniqueness of its representation.. In more general domains and/or for more general cone fields, this can be an issue. However, for the case considered here, the matrix will turn out to be invertible, so there is no ambiguity. Our assumption then is: Assumption C: The matrix ( =identity) is non-negative with spectral radius ". We refer to this as the contractive case for obvious reasons. We then have the following theorem: Theorem 4. Given a matrix satisfying assumption C, the SRP on with reflection matrix has a unique solution. The idea of [Harrison and Reiman 1981] and [Reiman 1984] is to consider an equivalent problem. The proof presented here differs slightly from their proof in that may be a discontinuous function. Also, existence and uniqueness are proved without reference to the alternative formulation of the SRP (see Definition 4) below. Proof. First consider a trivial instance of the problem, namely the SRP( ) (i.e. take the reflection matrix to be identity). Then the problem is a collection of 1-dimensional problems, and hence existence and uniqueness is immediate coordinate by coordinate. Let. denote this solution, when the free process is. The index refers to the identity reflection matrix. We also set 34. The results of Section 3 show that and are continuous both in the and topologies. Now consider satisfying Assumption C and set. Our equation 4. (see (i) of Definition 3) can be written as and since (ii) should hold, it follows that " ".. (12) and this is a fixed point equation. Since ", the inverse of is non-negative matrix. With denoting the all ones column, let, and is a

8 Notice that has strictly positive components. Instead of the usual norm on, we introduce the -norm defined by: This induces a norm on the matrices, which is easily seen to be: With this norm, we have " &, i.e. To see this, observe that & for all, or, $" for all. Thus, is a contractive linear function on ". (It was to be expected that is contractive under some norm, because ", but what we show here is that it is so under the - norm, and this is of relevance.) We also introduce the corresponding uniform norm on - ) &*+,, as in (8), and the metric, as in (10): " #! & # - ) &*+,2 The first one generates the -topology. The second one generates the complete and " #. It now follows, from (7) and the definition of, that: "! "!# "! "! Fix now a free motion matrix norm yield: Thus ) ) # - ) &*+,. The above inequalities and the fact that 2". 2". ". 8 topology. They are both is an induced ". ". ". 2".... is a contraction under the norm. Let be the operator applied times to a fixed / ( is thus Cauchy under the norm. Hence it metric. But this metric is complete and generates the topology. Hence -converges to, say,.. Now is -continuous and hence -continuous. Thus ) &*+. The sequence is Cauchy under the converges to.. But 2 and this of course converges to.. Thus. 2". and this proves that the limit point. solves the SRP( ). If there are two solutions, then the previous

9 9 inequality shows that they must be identical. So there is a unique solution. As in dimension one, we have another problem: Definition 4 (Monotonic formulation of the SRP). Given a matrix and, - ) &*+,, consider the set. - ) &*+, 4. $,&(, equipped with the componentwise partial ordering. A solution to the SRP( ) is a minimal element of this set. The equivalence of the two definitions needs a proof. Theorem 5. Assume that condition C holds. Then the SRP( ) of Definition 4 has a unique solution which coincides with the solution of the SRP( ) of Definition 3. Proof. We must show that any. - ) &*+,, such that 24. $ &, satisfies.$., where. is the unique solution of the SRP( ) in its standard formulation. Recall the definition of from (12) through which is is componentwise defined as in (5). It is then clear that is increasing in the natural partial order of functions: For all. #. - ) &*+,.6... Fix. - ) &*+,, such that 24.$,&. Thus ". 14. $,&. From Theorem 2 we obtain and so, by induction,. $ "... $. for all $ We now show that we can pass to the limit in this inequality. Recall from the proof of Theorem 4 that. -converges to.. Thus, there is a sequence such that -converges to id, and. -converges to. ; in particular, the convergences hold pointwise; the inequality in the display above also holds pointwise. Hence. "!##,.1 "!## for all! and. Taking limits we get. "!# #.1 "!#. But, given any,&, we can find such that "!! 4! for all $. Thus, for all $ we have.1 "!##,."! 4. Hence #.1 "!##,."!, by the right-continuity of.. Thus, in the end, we get This concludes the proof.. "!6.1"!# for all! $,&. 5 Behavior under weaker topologies Even though the the notion of convergence. We introduce the - ) &*+,, let topology is standard in the space - ) &*+,, it is often necessary to weaken " "!# "!##! $,&( topology, defined in [Skorokhod 1956]. Given

10 10 be the graph of, and let "1 "! 5 "! 4 "!##!$&( (13) be the linear extension of the graph of. Both sets are subsets of ) &*+,. Definition 5. An where the first coordinate -parametrization of an element of - ) &*+, is a bijection &, while the second coordinate is simply continuous, i.e. ) &*+,. ) &*+ is continuous, increasing 5, with Definition 6. We say that a sequence - ) &*+ exist -parametrizations of, (, and -converges to. " & -converges to an,6- ) &*+, if there -parametrization of, such that Note that this is a well-defined notion of convergence on - ) &*+, that leads to the usual definition of closed sets. The topology can be seen to be strictly weaker than : for instance, if we consider "! 1"! $ and "!# "! &, 6 (, we see that there is no way for to -converge to, but that does, indeed, -converge to ; see Figure 1. 1 x x n 1 1+1/n t Figure 1: converges in to, but not in 5.1 -continuity in dimension one We specialize to dimension first. The following propositions and lemma are responsible for the -continuity of the reflection mapping. Note that for an increasing function we let "! $,&!! $,&* be its generalized (right-continuous) inverse function. We also let "! "! $,& $!! $,& 5 but by no means strictly increasing in general

11 11 In the sequel we fix - ) &*+ and consider an notation, we set It is helpful to recall that "! # "!# ranges in in " as ranges over ) &*+. -parametrization 5 2 Proposition 1. Consider - ) &*+ and let 5 be an Proof. Recall that 3 1 2) &*+,. To ease the " as! ranges over ) &*+,, while # ranges -parametrization. Then (14) (15) "1 is a bijection. Hence any point in "1 has a unique pre-image in ) &*+,. In particular, consider the point ## for an arbitrary $&. This ## actually belongs to "1 which is contained in ". Hence it has a unique pre-image under. Call this pre-image. (See Figure 2.) Thus, * ##. x x trajectory x ( λ (u)) p(u) 0 t 0 λ(u) t u λ 1 ( λ(u) ) = v λ trajectory u Figure 2: Behavior of the -component of an function parametrization around a typical jump of a typical

12 12 But * *#, and so * # ##2 (16) Equating the first components of (16) we have, and it is easily seen that # #. Equating the second components of (16) we have * #. Substituting for we have # ## # and this holds for an arbitrary. But id, because is continuous. Hence, setting "! in the above display, we get "!# "! and this proves (14). To prove (15), i.e. that "! #% "!, for any!, we observe that # for some. So we set in place of in the last display, take limits, and argue similarly. Proposition 2. Under the assumptions and notation above, we have: Proof. Owing to proposition 1, we have to prove or that (see eqrefname2) (17) (18) # " for all! $&. But this follows from the fact that is strictly increasing. The second equality is proved similarly. Lemma 2. Consider - ) &*+, and let 1 be an 5 is an -parametrization for. -parametrization for. Then Proof. We only need to prove that. ' 5 1 is a bijection from ) &*+, into the linear extension of the graph of. By the assumption that 5 is an -parametrization for, we have that is a bijection from ) &*+, into ". So it suffices to show that there is a bijection "1 1 such that.2, or that the following diagram commutes: ) &*+ ) &*+ " 1 The function is defined in the obvious way: "! 5 "! 4 "!# "! "! 4 1 "!##! $ &* & % (19)

13 To show that it is one-to-one, take two distinct points in 13 " and show that the images are distinct. We only need consider distinct points of the form "! "! 4 "!## and "! "! 4 "!## for. If their -images were equal we would have 1 "! 4 "!# "! 4 1 "!#, and this means that 1 "! 1 "!# which means that "! "! (by the very definition of ). But then "! "! 4 "!# "!# "!## "! "! 4 "!##, and this contradicts the assumption that the points are distinct. It is equally clear that covers 1. So is a bijection. We now check that the range of To check that this holds we use the definition (19) of and verify that the second coordinates are equal by applying (17) and (18). Theorem 6 ( -continuity in one dimension). 1- ) &*+ - ) &*+, is continuous when both domain and codomain are given the Proof. Suppose that is an -topology. convergent sequence with limit. Thus there are -parametrizations is -continuous, of, and 5 of, such that -converges to 5. Since we have that -converges to 1. On the other hand, due to Lemma 2, is an -limit the function. -parametrization of and 1 is an Now notice that, unfortunately, - ) &*+, is not a topological vector space [Whitt 1980] be- continuous, as the following little example shows; see Figure 3. Let convergent to cause addition is not -parametrization of. Hence has as "!# 1! &, "! "!, (. Both sequences are 1"!$,& and 1"! $,&, respectively. We would expect that their sum converges to the sum of their limits which is zero. But their sum is not -convergent. So continuity of cannot be deduced from the continuity of. It must be considered separately. It is easily verified that Propositions 1, 2 and Lemma 2 remain true with in place of. Thus Theorem 6 is also true with in place of, showing that is also continuous in dimension one. 5.2 Dimension two and higher In this section we show, by examples, that not everything is fine with the $. topology in dimension The following observation was made by [Lin 1996]. There is something unnatural about the way that jumps are connected in the topology. Namely, it is required that an -parametrization of a function ' - ) &*+, connect "! and "! by a straight line; see (13) and Definition 6. Unfortunately though, the mapping does not necessarily map straight segments into straight segments. This geometrical observation is the reason behind the fact that is not -continuous for $. To put it in another way, look at the natural map of (19) (with in place of ). The image of "1 under is not in general. For example, consider the SRP( ) in dimension with the trivial reflection matrix & &

14 !!!!! 14 1 x n 0 1/n t 1 x n 1 x n + x n 0 1/n 2/n t Figure 3: The sum of the and consider the function "! " where & "! "! "!# "!# limits of, is zero; however, 4 does not converge in #"!*!!! #"!!!!! $ &* (20) ( until time!, " ; see Figure 4. This function remains at the point moves linearly from to 77&* between! and!, and remains at thereafter. Since is identity, we obtain its reflection by reflecting each component individually: The straight segment joining straight segments ) and "! " "!# "! 2 (21) by is bent by the reflection and becomes the union of the and ) &, where 37 &. How this specifically affects the continuity of at the point can be exhibited by choosing! and! so close to a fixed time, say, so that "! -approximates a function that jumps from to at time. Specifically, let!, that "!#, let "! be the function given by (20), and "! its reflection as in (21). It is clear -converges to the function "!# 1"! " 4 1"! $. The reflection of is 1"! ". However, does not converge in at all. It is easy then to conjecture that is not the right topology for. A weaker topology in its codomain or a stronger topology in its domain, that takes into account the special nature of, is required. Under such a modification one hopes that will become continuous. The problem is to find a modification that also has a number of other interesting properties (preserves continuity of certain

15 15 x 2 r 1 A = (2,1) graph of z r 2 0 C= (1,0) x 1 (0, 1) = B graph of x Figure 4: Straight-segment graphs can be bent when reflected in 2 or more dimensions popular functionals, for example). While this article was being written, [Whitt 1999b] proposed an interesting modification which makes the mapping continuous. Loosely speaking, a weaker topology is obtained by allowing jumps to be connected, not necessarily by straight segments, but continuous segments constrained in a box; for details we refer to the preprint [Whitt 1999a]. 5.3 A very weak topology Finally, we discuss the topology. Again this was introduced by [Skorokhod 1956], where its exact definition can be found. is considerably weaker than because it ignores the! -dependence of functions and only looks at their graphs. Loosely speaking, two functions # are -close if their graphs are uniformly close. It was observed in [Lin 1996] that if we endow both domain and codomain of with, then is discontinuous even in dimension one. This is shown by the following example; see Figure 5: Let be defined by linearly interpolating between the points & &,, &,, and "! for all! $. Then - converges to "!# 1"!%$. Now, with,, we have: is obtained by linearly interpolating between the points & &,, &, and "!# & for all!$ ; on the other hand, "! &. It is clear that the graph of is not close to the graph of. Hence does not -converge to. References [Billingsley 1968] Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.

16 16 1 z n 0 1 3/n 1 t 1 x n 1/n 1/n 1/n Figure 5: converges in to 1"!$, but does not converge in [Harrison and Reiman 1981] Harrison, J.M. and Reiman, M.I. (1981). Reflected Brownian motion on an orthant. Ann. Probab. 9, [Lin 1996] Lin, S.-J. (1996). Lévy and fractional Brownian motions for modeling and performance analysis in high-speed communication networks. Ph.D. dissertation, Univ. of Texas at Austin. [Reiman 1984] Reiman, M.I. (1984). Open queueing networks in heavy traffic. Math. Oper. Res. 9, [Skorokhod 1961] Skorokhod, A.D. (1961). Stochastic equations for diffusions in a bounded region. Theory Probab. Appl. 6, [Skorokhod 1956] Skorokhod (1956). Limit theorems for stochastic processes. Theory Probab. Appl. 1, [Tanaka 1979] Tanaka, H. (1979). Stochastic differential equations with reflecting boundary conditions in convex regions. Hiroshima Math. J. 9, [Whitt 1980] Whitt, W. (1980). Some useful functions for functional limit theorems. Math. Oper. Res 5, [Whitt 1999a] Whitt, W. (1999a). On the Skorohod topologies. Preprint. [Whitt 1999b] Whitt, W. (1999b). The reflection map is Lipschitz with appropriate Skorohod metrics. Preprint.